Ultra-confined Propagating Exciton–Plasmon Polaritons Enabled by Cavity-Free Strong Coupling: Beating Plasmonic Trade-Offs

Hybrid coupling systems consisting of transition metal dichalcogenides (TMD) and plasmonic nanostructures have emerged as a promising platform to explore exciton–plasmon polaritons. However, the requisite cavity/resonator for strong coupling introduces extra complexities and challenges for waveguiding applications. Alternatively, plasmonic nano-waveguides can also be utilized to provide a non-resonant approach for strong coupling, while their utility is limited by the plasmonic confinement-loss and confinement-momentum trade-offs. Here, based on a cavity-free approach, we overcome these constraints by theoretically strong coupling of a monolayer TMD to a single metal nanowire, generating ultra-confined propagating exciton–plasmon polaritons (PEPPs) that beat the plasmonic trade-offs. By leveraging strong-coupling-induced reformations in energy distribution and combining favorable properties of surface plasmon polaritons (SPPs) and excitons, the generated PEPPs feature ultra-deep subwavelength confinement (down to 1-nm level with mode areas ~ 10–4 of λ2), long propagation length (up to ~ 60 µm), tunable dispersion with versatile mode characters (SPP- and exciton-like mode characters), and small momentum mismatch to free-space photons. With the capability to overcome the trade-offs of SPPs and the compatibility for waveguiding applications, our theoretical results suggest an attractive guided-wave platform to manipulate exciton–plasmon interactions at the ultra-deep subwavelength scale, opening new horizons for waveguiding nano-polaritonic components and devices. Supplementary Information The online version contains supplementary material available at 10.1186/s11671-022-03748-7.


Numerical methods and equations for characterization
The permittivity of the monolayer WS 2 is described by a Lorentz oscillator model: with parameters from Ref. [2], where ε b,Ag =3.7, ħω p,Ag =8.9 eV and ħγ Ag =20 meV.
For the PEPP in our coupling system, it can be obtained by numerically solving the eigenvalue problem of the wave equation with the time-harmonic field given by E(x, y, z, t) = E(x, y)e i (kz-ωt) , where µ and ε are permeability and permittivity, (x, y, z) represents the spatial coordinates indicated in Fig. 1, ω and k are the angular frequency and the wavevector parallel with the MNW, respectively. The solutions can either be complex ω with real k or complex k with real ω [3]. For the complex-ω solution, the eigenvalue of ω is solved by feeding the eigenequation with a real-valued k, where the real part (Re(ω)) and the imaginary part (Im(ω)) of the ω represent the eigenfrequency and the temporal damping of the PEPP, respectively. As to the complex-k solution, the eigenvalue of k is solved by feeding in a realvalue ω, where the real part (Re(k)) and the imaginary part (Im(k)) of the k correspond to the propagation constant and the spatial damping along the propagation direction.
where the subscript indicates the integration area (m=MNW, l=WS 2 , tot=total region) and W(x, y) is the energy density. And for the dispersive and absorptive material, the energy density is calculated by part Re(ε), imaginary part Im(ε), and damping frequency γ of the complex permittivity [4]. For characterization of the spatial confinement and loss relation, the mode area (A m ) is defined as [5] ( , ) And the loss (damping along the z direction) is inversely proportional to the propagation length (L m ), which is calculated through the imaginary part of the complex-valued k [6]

Parameters for the coupled-oscillator (COM) model
The PEPP can be regarded as the hybridization of SPPs and excitons. In COM model, the eigen frequency of the PEPP can be obtained via the diagonalization of the Hamiltonian [7] Here ħω ex =2.013 eV and ħγ ex =22 meV are the exciton resonance energy and damping of the WS 2 material from the Lorentz model [1]. ω SPP =Re(ω) and γ SPP =2|Im(ω)| are the eigen frequency and damping frequency of the SPP mode, which are obtained from the complex-ω solution of the SPP mode. ħΩ R /2 represents the coupling strength at the zero detuning (ω SPP= ω ex ). For the parameters used in Fig. 5a(ii) in the main text, Figs. S1-S2 give the numerically obtained ħω SPP and ħγ SPP of the SPP (using the method in section1) with the varied MNW diameters (D) from 75 to 400 nm. For reference, the corresponding exciton counterparts ħω ex and ħγ ex are also plotted as black lines.  From the Fig. S1, we can also obtain the k at zero detuning (the cross point of the orange and black line where ω SPP = ω ex ). By feeding the k at the zero detuning into the eigenequation for the PEPPs, we can obtain the eigenvalues of the upper polaritons and lower polaritons, yielding ħΩ R listed in the Table S1. With the parameters shown in Figs. S1-S2 and Table S1, the dispersion relation in terms of ħRe(ω) vs. k of the PEPP for MNWs with D from 75 to 400 nm in Fig. 5a(ii) can be obtained using the COM model. For verification, Fig. S3 gives the comparison with D = 100 nm between the one obtained using COM model and the one from the simulation, showing excellent agreement with each other.

3D simulations
For 3D simulations, the geometrical configuration with Cartesian coordinates is shown in Fig. S11, where a MNW with the cross section in the x-y plane is placed along the z axis (propagation direction). The simulation region is firstly discretized into a triangular mesh with a minimum element size of 0.2 nm at the transverse cross section, and swept along the propagation direction. The length of the MNW (L) is set to be 1 µm with two numeric ports [8] deployed at the left-side-and right-side-boundaries. The left port is used as an input for exciting the waveguide, while the right one is used as an exit to detect and absorb the outgoing wave.
For the rest of the boundaries, perfectly matched layer (PML) conditions are used. To verify the 3D model, Fig. S12 gives the calculated energy density distributions along the propagation direction of our proposed WS 2 -clad MNW working at different wavelengths.
Since the energy is highly concentrated in the 1-nm WS 2 cladding and the features in Fig.   S12a(i-iii) are difficult to distinguish, we further normalize them and plot in a color bar with saturation [9] for better visualization (Fig. S12b(i-iii)). As are shown, when the wavelength approaches the excitonic resonance (616 nm), the loss of the PEPP dramatically increases, resulting in a sharply reduced propagation length (L m ), which coincides very well with our theoretical prediction in the manuscript. For further quantitative validation, we calculate L m from the transmission S 21 parameter between the output and input ports [8] Fig. 4(b)), a good agreement between them can be achieved.

Extending the strong coupling strategy to other structures
Besides the WS 2 -clad cylindrical MNW we demonstrated in the manuscript, the cavityfree strong coupling strategy can be further extended to other configurations such as a WS 2clad pentagonal MNW (Fig. S13) and a bare MNW on a flat substrate with a WS 2 layer on top of it (Fig. S14), enabling large Rabi splitting (Fig. S13(a) and Fig. S14(a)) and tightly confined energy inside the WS 2 layer (Fig. S13(b) and Fig. S14(b)).